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These axioms are based on Whitehead, "The Axioms of Projective Geometry". There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are: G1: Every line contains at least 3 points; G2: Every two distinct points, A and B, lie on a unique line, AB.
In synthetic geometry, point and line are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the ...
This famous incidence geometry was developed by the Italian mathematician Gino Fano. In his work [9] on proving the independence of the set of axioms for projective n-space that he developed, [10] he produced a finite three-dimensional space with 15 points, 35 lines and 15 planes, in which each line had only three points on it. [11]
A projective space S can be defined abstractly as a set P (the set of points), together with a set L of subsets of P (the set of lines), satisfying these axioms : . Each two distinct points p and q are in exactly one line.
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language ( § Principle of duality ) and the other a more functional approach through special ...
For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory (with more models) than is found by starting with a vector space of dimension three. Projective geometry has in fact the simplest and most elegant synthetic expression of any geometry. [5]
In projective geometry, Desargues's theorem, named after Girard Desargues, states: Two triangles are in perspective axially if and only if they are in perspective centrally . Denote the three vertices of one triangle by a , b and c , and those of the other by A , B and C .
Since it is a projective space, algebraic techniques can also be effective tools in its study. In a separate usage, a Fano plane is a projective plane that never satisfies Fano's axiom; in other words, the diagonal points of a complete quadrangle are always collinear. [1] "The" Fano plane of 7 points and lines is "a" Fano plane.